Historical Context & Motivation
Long before calculators existed, astronomers and navigators needed ways to compute trigonometric values for angles that didn't appear in standard tables. The angle addition and subtraction formulas arose from this practical need: how do you find sin 75° if you only know sine and cosine for 45° and 30°? The answer lies in breaking a complicated angle into the sum or difference of simpler ones. This idea stretches back thousands of years and remains one of the most powerful tools in trigonometry.
The central question these formulas answer is deceptively simple: if you know the sine and cosine of two individual angles α and β, can you find the exact sine, cosine, and tangent of their sum α + β or difference α − β without measuring the combined angle directly? The answer is a definitive yes, and proving why these formulas work is the focus of this lesson.
Core Principles & Definitions
Before diving into proofs, you need a firm grasp of a few foundational ideas. These principles tie together the unit circle, distance formula, and the special relationship between sine and cosine that makes the addition formulas possible.
Unit Circle Coordinates
The Pythagorean Identity
The Distance Formula
Cofunction & Even/Odd Identities
Visual Proof: The Unit Circle Approach
The most intuitive proof of the cosine subtraction formula uses the unit circle and the distance formula. The diagram below shows two configurations of the same arc on the unit circle. By rotating the arc so that one endpoint sits on the positive x-axis, we can set up two distance expressions that must be equal — and expanding them reveals the formula.
Here is the algebra. In Configuration 1, use the distance formula between P = (cos α, sin α) and Q = (cos β, sin β): d₁² = (cos β − cos α)² + (sin β − sin α)². Expand this to get cos²β − 2 cos α cos β + cos²α + sin²β − 2 sin α sin β + sin²α. Apply the Pythagorean identity (cos²θ + sin²θ = 1) twice to simplify: d₁² = 2 − 2(cos α cos β + sin α sin β).
In Configuration 2, the distance from A = (1, 0) to B = (cos(β − α), sin(β − α)) gives d₂² = (cos(β − α) − 1)² + (sin(β − α))². Expanding: d₂² = cos²(β − α) − 2 cos(β − α) + 1 + sin²(β − α) = 2 − 2 cos(β − α). Setting d₁² = d₂² yields the cosine subtraction formula: cos(β − α) = cos α cos β + sin α sin β.
Mathematical Framework: All Six Formulas
Once you have the cosine subtraction formula, every other addition/subtraction formula can be derived from it using only the even/odd identities and the cofunction identities. Let's build them one by one.
Tangent Formulas
The tangent addition and subtraction formulas follow from dividing the sine formula by the cosine formula, since tan θ = sin θ / cos θ. After dividing both numerator and denominator by cos α cos β, you get a clean result.
Step-by-Step Derivation Map
The diagram below illustrates the logical flow from the single master identity — cos(α − β) — to all six addition/subtraction formulas. Each arrow represents one algebraic substitution.
| Formula | Expression | Key Derivation Step |
|---|---|---|
| cos(α − β) | cos α cos β + sin α sin β | Unit circle distance proof (master identity) |
| cos(α + β) | cos α cos β − sin α sin β | Substitute −β for β; use cos(−β) = cos β, sin(−β) = −sin β |
| sin(α − β) | sin α cos β − cos α sin β | Cofunction: sin θ = cos(π/2 − θ) |
| sin(α + β) | sin α cos β + cos α sin β | Substitute −β for β in sine subtraction |
| tan(α + β) | (tan α + tan β) / (1 − tan α tan β) | Divide sin(α + β) by cos(α + β); simplify |
| tan(α − β) | (tan α − tan β) / (1 + tan α tan β) | Substitute −β for β in tangent addition |
Worked Example: Finding sin 75° Exactly
Let's put the sine addition formula to work. We want to find the exact value of sin 75°. Notice that 75° = 45° + 30°, and we know exact values for sine and cosine of both 45° and 30°.
Common Mistakes & How to Avoid Them
When working with angle addition formulas, a few errors come up again and again. The table below catalogs the most common mistakes alongside correct approaches, so you can learn to spot them before they trip you up.
| Common Mistake | Why It's Wrong | Correct Approach |
|---|---|---|
| sin(α + β) = sin α + sin β | Sine is NOT a linear function. You cannot distribute it over addition like multiplication. | Use sin(α + β) = sin α cos β + cos α sin β. |
| Mixing up signs in the cosine formula | "Cosine is contrary": the sign between terms is opposite to the operation. cos(A + B) uses minus, not plus. | cos(A + B) = cos A cos B − sin A sin B. Remember: addition → minus. |
| Forgetting to use exact special-angle values | Decimal approximations lose exactness and defeat the purpose of these formulas. | Keep radicals: sin 45° = √2/2, cos 30° = √3/2, etc. |
| Dividing wrong when deriving tangent | You must divide BOTH numerator and denominator by cos α cos β, not just the top. | tan(α + β) = sin(α + β)/cos(α + β), then divide every term by cos α cos β. |
Connection to Advanced Identities
The angle addition formulas are not an isolated topic — they are the foundation for a whole family of more advanced trigonometric identities that you'll encounter in later courses. Understanding this connection shows you why mastering these formulas now pays dividends later.
| Addition Formulas (This Lesson) | Advanced Identity | How It Connects |
|---|---|---|
| sin(α + β) = sin α cos β + cos α sin β | Double-Angle: sin 2θ = 2 sin θ cos θ | Set α = β = θ in the sine addition formula |
| cos(α + β) = cos α cos β − sin α sin β | Double-Angle: cos 2θ = cos²θ − sin²θ | Set α = β = θ in the cosine addition formula |
| cos(α − β) and cos(α + β) | Product-to-Sum: cos α cos β = ½[cos(α − β) + cos(α + β)] | Add the two cosine formulas and solve for the product |
| cos 2θ = 1 − 2 sin²θ | Half-Angle: sin²(θ/2) = (1 − cos θ)/2 | Solve the double-angle formula for sin²θ, then replace θ with θ/2 |
| Euler: eiθ = cos θ + i sin θ | Complex Exponentials | Multiplying eiα × eiβ = ei(α+β) reproduces both addition formulas at once |
In AP Calculus, you'll use these identities to differentiate and integrate trigonometric functions. In physics, the addition formulas describe how waves combine — the principle of superposition depends on exactly these relationships. If you continue to engineering or computer science, signal processing relies on them daily. Mastering the proofs now gives you a deep understanding that memorization alone can't provide.
Practice Problems
Lesson Summary
In this lesson you learned how to prove and apply the angle addition and subtraction formulas for sine, cosine, and tangent. The proof begins with the cosine subtraction formula, derived by comparing chord lengths on the unit circle using the distance formula. From this single master identity, every other formula follows through systematic substitutions: replacing β with −β using even/odd identities, applying cofunction identities to move between cosine and sine, and dividing to derive the tangent versions.
The six formulas are: cos(α ± β) = cos α cos β ∓ sin α sin β, sin(α ± β) = sin α cos β ± cos α sin β, and tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β). These identities let you compute exact trig values for non-standard angles like 15°, 75°, and 105° by decomposing them into sums or differences of special angles (30°, 45°, 60°). They also serve as the foundation for double-angle, half-angle, and product-to-sum identities that you'll encounter in more advanced courses.